A143508 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^2)^2.

1, 1, 2, 9, 52, 372, 3058, 28074, 282028, 3059328, 35497672, 437499541, 5696752234, 78036803430, 1120687989348, 16823652188164, 263345788211608, 4289062071449610, 72543038644585822, 1271980596430351862, 23085579883157411532, 433071407705851089244

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

Given g.f. A(x), let G(x) be defined by G(x*A(x)^2) = x, then

(1) G(x) = x/(1 + A(x)^2*G(x))^2,

(2) A(G(x)) = 1 + A(x)^2*G(x).

From Seiichi Manyama, Mar 01 2025: (Start)

Let a(n,k) = [x^n] A(x)^k.

a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(2*n-2*j+k,j)/(2*n-2*j+k) * a(n-j,2*j). (End)

Example

G.f. A(x) = 1 + x + 2*x^2 + 9*x^3 + 52*x^4 + 372*x^5 + 3058*x^6 +...
A(x)^2 = 1 + 2*x + 5*x^2 + 22*x^3 + 126*x^4 + 884*x^5 + 7149*x^6 +...
A(x*A(x)^2) = 1 + x + 4*x^2 + 22*x^3 + 156*x^4 + 1285*x^5 + 11886*x^6 +...
A(x*A(x)^2)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 372*x^4 + 3058*x^5 +...
Define G(x) by G(x*A(x)^2) = x, then
G(x) = x - 2*x^2 + 3*x^3 - 12*x^4 + 17*x^5 - 198*x^6 - 345*x^7 +...
such that G(x) = x/(1 + A(x)^2*G(x))^2.

Prog

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A^2, x, x*A^2)); polcoeff(A, n)}
(PARI) a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+k, j)/(2*n-2*j+k)*a(n-j, 2*j))); \\ Seiichi Manyama, Mar 01 2025

Crossrefs

Cf. A212029, A143501, A381568.

Sequence in context: A231494 A006152 A369551 * A052882 A248440 A330200

Adjacent sequences: A143505 A143506 A143507 * A143509 A143510 A143511

Keyword

nonn,changed

Author

Paul D. Hanna, Aug 21 2008

Status

approved